Blow-up rate and uniqueness of singular radial solutions for a class of quasi-linear elliptic equations

We establish the uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem -δ_pu=λu^p-1-b(x)h(u) in B_R(x₀) with boundary condition u=+∞ on B_R(x₀), where B_R(x₀) is a ball centered at x0∈RN with radius R, N≥3, 2≤p<∞, λ>0 are constants and the weight function b is a positive radially symmetrical function. We only require h(u) to be a locally Lipschitz function with h(u)/u^p-1 increasing on (0, ∞) and h(u)~u^q-1 for large u with q>p-1. Our results extend the previous work [Z. Xie, Uniqueness and blow-up rate of large solutions for elliptic equation -δu=λu-b(x)h(u), J. Differential Equations 247 (2009) 344-363] from case p=2 to case 2≤p<∞.